MinT - Best Known (57, 57+171, s)-Nets in Base 4

Best Known (57, 57+171, s)-Nets in Base 4

(57, 57+171, 66)-Net over F₄ — Constructive and digital

Digital (57, 228, 66)-net over F₄, using

t-expansion [i] based on digital (49, 228, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(57, 57+171, 91)-Net over F₄ — Digital

Digital (57, 228, 91)-net over F₄, using

t-expansion [i] based on digital (50, 228, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(57, 57+171, 243)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 228, 244)-net over F₄, because

3 times m-reduction [i] would yield digital (57, 225, 244)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁵, 244, F₄, 168) (dual of [244, 19, 169]-code), but
  - residual code [i] would yield OA(4⁵⁷, 75, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 175541 891167 451460 543215 010075 896682 965259 780096 / 6 746458 181029 > 4⁵⁷ [i]

(57, 57+171, 372)-Net in Base 4 — Upper bound on s

There is no (57, 228, 373)-net in base 4, because

1 times m-reduction [i] would yield (57, 227, 373)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 55687 338773 773127 411047 007682 101436 424600 498840 013246 845463 159526 736170 753575 702844 978955 931165 247087 939106 616889 535146 821974 675808 825956 > 4²²⁷ [i]